We introduce the norms in the Hardy spaces of martingales defined in [1]} and [2]}. Let \(1 \le p \le \infty \)  and \(x=\left(x_{n}\right)_{n \ge 1}\)  be an \(L_{p}\) -martingale. Set, for \(p \ge 2\)
\(\Vert x\Vert _{\mathcal {H}^{p}}=\max \left\lbrace \left\Vert S_c(x)\right\Vert _{p},\left\Vert S_r(x)\right\Vert _{p}\right\rbrace \) 
